The present invention relates to digital filters, and specifically to second-order digital filters. It is well known that any order of digital filters may be efficiently constructed from second-order building blocks.
Direct mechanization of digital filters is well understood by those skilled in the digital-filter art. The direct mechanization of a second-order digital filter requires five multipliers. Each multiplier used in the direct mechanization of the filter introduces round-off noise into the result because only a finite number of bits of the product can be carried forward. In addition, coefficient error is introduced because each multiplier coefficient is a finite number, having a finite number of bits. Therefore, the poles and zeros of the filter implemented in the direct form are not exact.
The normal-form of the second-order digital filter provides a digital filter structure with lower round-off noise, minimum coefficient inaccuracy errors, and an absence of limit cycles. Such a normal-form filter is described in the article by C. W. Barnes, "A Parametric Approach to the Realization of Second Order Digital Filter Sections", IEEE Trans. on Circuit and System, Vol. CAS-32 No. 6, June 1985, pages 530-539.
The normal-form filter provides better coefficient error and round-off error because inherent error correction is provided by the cross-coupling of the partial-product outputs within the filter.
A block diagram of the conventional second-order normal-form digital filter structure is shown in FIG. 1. The filter has an input 11 for receiving an input signal X, which may consist of a series of N-bit signal samples, x.sub.n. The signal is processed through a plurality of multipliers to produce intermediate results U and V the final result Y. The final result Y is placed on the filter output as the output signal. The intermediate results U and V are fed back and cross-coupled within the filter. The multipliers b.sub.1, b.sub.2, c.sub.1, and c.sub.2 determine the locations of the poles of the filter. The input multipliers a.sub.1 and a.sub.2 determine the input scaling. Once the other coefficients have been set, the multipliers a.sub.0, d.sub.1, and d.sub.2 determine the zero locations of the filter. The second-order normal-form filter has a total of nine multipliers. Element-by-element mechanizations of this filter, sometimes with variations as taught by Barnes, represents the prior art.
Nevertheless, the normal-form digital filter provides design difficulties for implementation and also is difficult to analyze. In addition, the large number of multipliers occupies a great deal of space on a semiconductor chip when this filter is implemented in microcircuitry. Therefore, efforts have been undertaken to reduce the number of multipliers in the second-order normal-form digital filter.